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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

On error estimates for Galerkin spectral discretizations of parabolic problems with nonsmooth initial data


Authors: Javier de Frutos and Rafael Muñoz-Sola
Journal: Math. Comp. 70 (2001), 525-531
MSC (2000): Primary 65M70, 65M15
Published electronically: March 1, 2000
MathSciNet review: 1680871
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Abstract:

We analyze the Legendre and Chebyshev spectral Galerkin semidiscretizations of a one dimensional homogeneous parabolic problem with nonconstant coefficients. We present error estimates for both smooth and nonsmooth data. In the Chebyshev case a limit in the order of approximation is established. On the contrary, in the Legendre case we find an arbitrary high order of convegence.


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  • 1. Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223. MR 0482275 (58 #2349)
  • 2. Christine Bernardi and Yvon Maday, Properties of some weighted Sobolev spaces and application to spectral approximations, SIAM J. Numer. Anal. 26 (1989), no. 4, 769–829 (English, with French summary). MR 1005511 (91c:46046), http://dx.doi.org/10.1137/0726045
  • 3. Claudio Canuto, M. Yousuff Hussaini, Alfio Quarteroni, and Thomas A. Zang, Spectral methods in fluid dynamics, Springer Series in Computational Physics, Springer-Verlag, New York, 1988. MR 917480 (89m:76004)
  • 4. C. Canuto and A. Quarteroni, Spectral and pseudospectral methods for parabolic problems with nonperiodic boundary conditions, Calcolo 18 (1981), no. 3, 197–217. MR 647825 (84h:35132), http://dx.doi.org/10.1007/BF02576357
  • 5. Robert Dautray and Jacques-Louis Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques. Tome 3, Collection du Commissariat à l’Énergie Atomique: Série Scientifique. [Collection of the Atomic Energy Commission: Science Series], Masson, Paris, 1985 (French). With the collaboration of Michel Artola, Claude Bardos, Michel Cessenat, Alain Kavenoky, Hélène Lanchon, Patrick Lascaux, Bertrand Mercier, Olivier Pironneau, Bruno Scheurer and Rémi Sentis. MR 902802 (88i:00003b)
  • 6. J. de Frutos and R. Muñoz Sola, Chebyshev pseudospectral collocation for parabolic problems with nonconstant coefficients, Proceedings of the third international conference on spectral and high order methods, Houston (Texas), 1996, 101-107.
  • 7. J. de Frutos and R. Muñoz Sola, Error estimates for Galerkin spectral discretizations of parabolic problems with nonsmooth data, Applied Mathematics and Computation Reports 1998/7, Universidad de Valladolid (Spain), pp. 749-754. CMP 98:15
  • 8. G. Fernández Manín. Algunas contribuciones al estudio del error en los métodos espectrales: optimalidad de los métodos de Jacobi y estudio del método de ``patching". PhD Thesis. Santiago de Compostela, 1995.
  • 9. A. Casal, L. Gavete, C. Conde, and J. Herranz (eds.), III Congreso de Matemática Aplicada/XIII C.E.D.Y.A. (Congreso de Ecuaciones Diferenciales y Aplicaciones), Universidad Politécnica de Madrid, Madrid, 1993 (Spanish). MR 1425638 (97g:00019)
  • 10. P. G. Ciarlet and J.-L. Lions (eds.), Handbook of numerical analysis. Vol. II, Handbook of Numerical Analysis, II, North-Holland, Amsterdam, 1991. Finite element methods. Part 1. MR 1115235 (92f:65001)
  • 11. Mitchell Luskin and Rolf Rannacher, On the smoothing property of the Galerkin method for parabolic equations, SIAM J. Numer. Anal. 19 (1982), no. 1, 93–113. MR 646596 (83c:65245), http://dx.doi.org/10.1137/0719003
  • 12. A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486 (85g:47061)
  • 13. Vidar Thomée, Galerkin finite element methods for parabolic problems, Lecture Notes in Mathematics, vol. 1054, Springer-Verlag, Berlin, 1984. MR 744045 (86k:65006)

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Additional Information

Javier de Frutos
Affiliation: Departamento de Matemática Aplicada y Computación, Universidad de Valladolid, Valladolid, Spain
Email: frutos@mac.cie.uva.es

Rafael Muñoz-Sola
Affiliation: Departamento de Matemática Aplicada, Universidad de Santiago de Compostela, Santiago de Compostela, Spain
Email: rafa@zmat.usc.es

DOI: http://dx.doi.org/10.1090/S0025-5718-00-01195-9
PII: S 0025-5718(00)01195-9
Keywords: Spectral Galerkin method, parabolic equation, nonsmooth initial data
Received by editor(s): January 4, 1999
Received by editor(s) in revised form: April 6, 1999
Published electronically: March 1, 2000
Additional Notes: J. de Frutos was partially supported by project DGICYT PB95-705 and project JCyL VA52/96. R. Muñoz-Sola was partially supported by project DGICYT PB96-0952.
Article copyright: © Copyright 2000 American Mathematical Society